 So we'll start looking at division algorithms by considering the basic approach to division, and this is division by subtraction. So again, we have our definition. If I have a multiplication, then I immediately have a division as well. And here's a useful idea to keep in mind. Notation in mathematics is never constructed at random. Good notation survives for a reason, and the reason usually is because it reminds us of something useful. So our Saint Andrew's cross that we use here for multiplication, the reason that that persists as a multiplication symbol is it reminds us that when you're doing a multiplication, what you're actually doing is a repeated addition. What about this thing? Well, this particular symbol that we have for division looks a lot like a subtraction, and that is not an accident. And the reason emerges as follows. If I have this multiplication a times b equal to c, then I know that c is the sum of a b's. So I'm going to add b to itself a whole bunch of times, and so by definition, if I have this product that will lead me to my division, then I also have this sum. And what do we know? Well, any time we have an addition, we also have a subtraction. So what I can also do is I can take c and subtract b a whole bunch of times, and eventually get down to zero. So this means that if I have a multiplication, I have a repeated addition, I have a repeated subtraction. And so I can rephrase the question. If I want to find what is c divided by b, the question I can answer instead is how many b's do I need to subtract before I get to zero? And this gives us division by repeated subtraction. So let's consider a very simple division problem by 36 divided by 12. So my question is going to be rephrased. How many 12's do I need to subtract before I can get to zero? So, well, let's just do it. So I'll start with 36, I'll subtract 12, I'll subtract 12, I'll subtract 12, and I'm at zero, and I've had to subtract 1, 2, 3, 12. So 36 divided by 12 is equal to 3. And I could actually do any division I want to this way. Of course, it may get a little bit less efficient. For example, consider 72 divided by 3. And I don't really want to sit here for the next half hour subtracting 3's. I could do it. It's not a thing that's going to be difficult. It's just going to take a while. But I can also instead subtract sets of 3's. So I don't have to subtract 3. I can subtract groups of 3. And the other thing I have to keep track of is how many sets of 3 I have subtracted. And just as an easy observation, 10 3's is 30. And so I might start by subtracting 10 3's all at once. So from 72, I subtract 30. That takes me down to 42. And I can do that again. Subtract another 30. And then now I can slow down and subtract 3's one at a time. And eventually I get down to zero. And if I count the number of 3's that I've subtracted, I'm going to get the quotient. Now let's see. So remember that's 10 3's there. Another 10. And then these are individual 3's. So I've subtracted 10, 20, 21, 22, 23. I subtracted a total of 24 3's all together. Now organization is really the big clue, the big issue here, because division is the most complicated of the basic operations. And part of it is it's one where the organization makes the most difference. So let's consider another way of organizing that subtraction. So maybe I'll organize it vertically. So here I've subtracted 30. And that's the same as 10 3's. I've subtracted another 30, another 10 3's. And then I've subtracted 1, 2, 3, 4 more 3's. And so I can organize the subtraction of these sets of 3 vertically. And the number of 3's that I've subtracted is just going to be the sum in this column. That's 10, 20, 21, 22, 23, 24. So I've subtracted 24 3's all together. And so my quotient, 72 divided by 3 equals 24.